High-order harmonic generation from a thin film crystal perturbed by a quasi-static terahertz field

Studies of laser-driven strong field processes subjected to a (quasi-)static field have been mainly confined to theory. Here we provide an experimental realization by introducing a bichromatic approach for high harmonic generation (HHG) in a dielectric that combines an intense 70 femtosecond duration mid-infrared driving field with a weak 2 picosecond period terahertz (THz) dressing field. We address the physics underlying the THz field induced static symmetry breaking and its consequences on the efficient production/suppression of even-/odd-order harmonics, and demonstrate the ability to probe the HHG dynamics via the modulation of the harmonic distribution. Moreover, we report a delay-dependent even-order harmonic frequency shift that is proportional to the time derivative of the THz field. This suggests a limitation of the static symmetry breaking interpretation and implies that the resultant attosecond bursts are aperiodic, thus providing a frequency domain probe of attosecond transients while opening opportunities in precise attosecond pulse shaping.


Comparison between the the parallel and perpendicular MIR-THz polarization configurations
In ref. [1] it has been shown that for bulk silicon (Si) crystal, there is a factor of 3 difference in the even-order harmonic conversion efficiency between the parallel and perpendicular MIR-THz polarization configurations, which agrees with perturbative calculation based on values of the χ (3) tensor. In our study, calculation based on χ (3) of ZnO suggests that there should be a factor of 10 difference, which deviates from the experimental observation We also find an interesting feature for the modulation of the total harmonic yield (Y tot ): When the MIR and THz fields are parallel, the generation/suppression of even-/odd-order harmonics are similar in amplitude (main text figure 2b), and Y tot always increases on the order of 1-2% (figure 3b, numbers). Whereas with orthogonally polarized MIR and THz fields, the suppression of odd-order harmonics is slightly stronger than the generation of even-orders (figure 2a, inset), and Y tot decreases on the order of 1% (figure 2b, numbers).
These are consistent with a generalized recollision HHG process: With the THz perturbing field parallel to the MIR drive, the total peak field slightly increases, leading to a very small increase in the excitation and therefore an enhancement of Y tot . Whereas with orthogonally polarized MIR and THz fields, the transverse displacement of the electron driven by the THz field reduces the electron-hole recollision probability, therefore Y tot is suppressed.
In ref. [2], the relative even-order harmonic yields from an orthogonal ω-2ω drive have been used to extract the HHG recollision angles. It was shown that (for short trajectories) electrons associate with lower-order harmonics experience larger recollision angles, as they are "born" when the fundamental field is weak and are affected more by the second harmonic perturbing field. Following the same argument, we calculate the recollision angles with orthogonal MIR-THz two-color field from single-shot measurement of the harmonic spectrum (figure 2b). Our results distinct from ref. [2] that the recollision angle increases with the harmonic order, implying that electrons associate with higher-order harmonics spend more time in "continuum" and drift more in the transverse direction due to the THz drive. This clearly indicate that the second harmonic and the THz perturbing fields fall in different Keldysh scaling regimes.

Probing the band structure via THz-dressed HHG
Our semiclassical analysis of the electron motion during the generalized recollision HHG process shows that the THz field induces a first-order perturbation to the electron dipole phase difference δϕ = ω(∆t c −0.5T 0 )−∆S ∝ F THz , and the odd-order harmonic yield follows a cos 2 (δϕ/2) suppression. Since the equation of motion of the electron is directed related to the reciprocal space group velocity (the band dispersion of the crystal), the band structure information is imprinted on the relationship between δϕ and the harmonic order.
To verify the ability to probe the band structure, we perform semiclassical calculations for the top three valence bands: heavy holes, light holes and split-off [3]. These three bands have very close energies at Γ point, but different band dispersions lead to different electron trajectories after the excitation. Figure 4 shows calculations of the THz induced harmonic yield modulation: For short trajectories, the split-off band deviates more and more from the heavy and light holes bands as the harmonic order increases. For long trajectories, the order-dependence of the modulation is very sensitive to the band structure.  As is shown, from the 5 th -to the 19 th -orders, all harmonic yields scale linearly with the THz intensity, this linear dependence could be utilized for precise THz metrology. In figure 6, we measure the individual harmonic yield as a function of the MIR-THz field delay, which shows that all harmonic orders are in phase, further supports that the harmonic yield modulation responds to the instantaneous THz intensity and sub-cycle dynamics have been washed out on the ps THz period time scale. We also compare the THz field sampled by electro-optic sampling (EOS), SHG, and HHG (figure 6c). Though we do not draw the conclusion from the comparison that the HHG method has better resolution, as our experimental setup is not optimized for EOS, we do provide the following arguments that the THz-dressed HHG has the potential for developing ultra-broadband, ultra-sensitive THz time domain spectroscopy (THz-TDS) [4]: Conventional EOS has bandwidth limit because the EO coefficient of the sampling crystal is frequency dependent and may vary a lot within the THz frequency range. For example, ZnTe is typically good for frequencies below 3 THz. If we use highorder harmonics to sample the THz field, such bandwidth limit is released. In addition, since the effective interaction length for (above-band gap) HHG is ultra-thin, typically tens of nm, walk-off effect due to velocity mismatching between the THz and the probe pulses is eliminated. Besides, the highly nonlinear nature of the HHG process effectively shortens the probe pulse duration, resulting in an increased temporal resolution. Last but not least, EOS typically uses crystals with large χ 2 to achieve strong Pockels effect, whereas in principle, any crystal can realize HHG based THz field sampling. However, since the harmonic yield directly samples the THz intensity, careful design is needed to determine the polarity of the THz field. For example, a DC bias can be applied, or χ 2 crystals that can generate even-order harmonics in the absence of the THz field can be used.

Even-order harmonic frequency shift: Extended study
To unravel more the delay-dependent even-order harmonic frequency shift, we perform SBEs simulations to investigate its sensitivity to the crystal band gap and the MIR characteristics: wavelength, short pulse carrier-envelope-phase (CEP) and group delay dispersion In the fitting equation, ω 2n /ω 0 = C 0 + C 1 * F ′ THz (t)/F THz (t), the frequency offset (C 0 − 2n) and delay-dependent shift (C 1 ) depend on the harmonic order. Interestingly, both parameters approach 0 and reverse sign at around the 14 th -order harmonic. SBEs simulations by varying the crystal band gap (while keeping the shape of the band curvature unaltered) suggest that the amplitude and sign of C 0 − 2n and C 1 are sensitive to the width of the band gap ( figure 15). This indicates that although the origin of the frequency offset and delay-dependence comes from the MIR field being pulsed and the THz field being non-DC, the crystal band gap also plays an important role in determining the exact value of the emitted harmonic frequencies. To the best of our knowledge, there has been no systematic study about the relationship between the crystal band gap/structure and the harmonic frequency on a fine level and open questions remain. In the future, experimental investigation can be performed with high resolution spectrometer and careful preparation of different crystal samples (e.g. thin films to rule out nonlinear distortion and propagation effects).
Theoretical verification may be further supported by more "complete" simulation beyond one-dimensional two-band model.
To investigate whether the ratio of the MIR to THz period/wavelength plays a role in the frequency shift, we compare SBEs simulations at two MIR wavelengths, λ = 2.4 and 3.6 µm, respectively. The results show a MIR wavelength-dependence of C 1 as a function of the harmonic order. However, if we compare at a fixed harmonic photon energy, C 1 only experiences small MIR wavelength-dependence (figure 16), perhaps suggesting that as long as the THz period is much longer than the MIR pulse duration, the frequency shift is not sensitive to the MIR period.
For strong-field light-matter interaction, it is known that as the driving laser pulse duration becomes very short, there will be CEP effects. In our SBEs simulations, we find that the CEP of the MIR pulse starts to affect the harmonic frequency and the fitting equation  Figure 19 compares the experimental measurement with the SBEs simulation the relationship between the MIR GDD and the even-order harmonic frequency shift.
Last but not least, we think more insights between the frequency shift and the HHG process may be unraveled via semiclassical analysis. However, such analysis is extremely challenging and beyond the scope of this work. In the following, we provide some attempt to show the relationship between the slope of the perturbing field and the frequency of the harmonic light. We assume a continuous wave MIR field and a linearly ramped perturbing field F l (t) = F 0 + F ′ * t, here F 0 is the DC offset and F ′ = dF l /dt is the ramp rate. Following similar analysis as is used in the METHOD section, within the first order approximation, we can define the recombination times (t c ) and the semiclassical actions (S) for the up ("+") and down ("−") trajectories of the m th MIR field cycle, here the linear-dependence (of t c and S on the THz field strength) parameters α, β depend on the harmonic order, and are positive/negative for short/long trajectories, respectively. We then separately calculate harmonics generated from the sum of all up or down trajectories, and, j ± inter (ω ± ) 2 ∝ sin N (ω ± ± ω ± αF ′ ∓ βF ′ )T 0 /2 N sin (ω ± ± ω ± αF ′ ∓ βF ′ )T 0 /2 2 (4) the q th "harmonic" frequency ω ± q must satisfy, (ω ± q ± ω ± q αF ′ ∓ βF ′ ) = qω 0 (5) therefore, For a cw-MIR drive, ω + q and ω − q are well separated, meaning that the harmonic frequency will be double-spitted with ∆ω q = ω + q −ω − q ∝ F ′ upon the application of the linearly ramped perturbing field. For a pulsed MIR drive, the harmonic frequency has certain bandwidth, and a quantitative semiclassical analysis becomes challenging. However, the interference between the up and down trajectories, which depends on both F 0 and F ′ , affects the final emitted harmonic frequency.